RECOVERING A TIME-HOMOGENEOUS STOCK PRICE
PROCESS FROM PERPETUAL OPTION PRICES
ERIK EKSTRÖM AND DAVID HOBSON
Abstract. It is well-known how to determine the price of perpetual
American options if the underlying stock price is a time-homogeneous
diffusion. In the present paper we consider the inverse problem, i.e.
given prices of perpetual American options for different strikes we show
how to construct a time-homogeneous model for the stock price which
reproduces the given option prices.
1. Introduction
In the classical Black-Scholes model, there is a one-to-one correspondence
between the price of an option and the volatility of the underlying stock.
If the volatility σ is assumed to be given, for example by estimation from
historical data, then the arbitrage free option price can be calculated using
the Black-Scholes formula. Conversely, if an option price is given, then the
implied volatility can be obtained as the unique σ that would produce this
option price if inserted in the Black-Scholes formula. It has been well documented that if the implied volatility is inferred from real market data for
option prices with the same maturity date but with different strike prices,
then typically a non-constant implied volatility is obtained. Since the implied volatility often resembles a smile if plotted against the strike price, this
phenomenon is referred to as the smile effect. The smile effect is one indication that the Black-Scholes assumption of normally distributed log-returns
is too simplistic.
A wealth of different stock price models has been proposed in order to
overcome the shortcomings of the standard Black-Scholes model, of which
the most popular are jump models and stochastic volatility models. Given a
model, option prices can be determined as risk neutral expectations. However, models are typically governed by a small number of parameters and
only in exceptional circumstances can they be calibrated to perfectly fit the
full range of options data.
Instead, there is a growing literature which tries to reverse the procedure
and uses option prices to make inferences about the underlying price process.
At one extreme models exist which take a price surface as the initial value
of a Markov process on a space of functions. In this way the Heath-JarrowMorton [6] interest rate models can be made to perfectly fit an initial term
structure. Such ideas inspired Dupire [5] to introduce the local volatility
model which calibrates perfectly to an initial volatility surface. For a local
Date: March 26, 2009.
1
2
ERIK EKSTRÖM AND DAVID HOBSON
volatility model, Dupire derived the PDE
1
PT (T, K) + rKPK (T, K) = σ 2 (T, K)K 2 PKK (T, K)
2
where P (T, K) is the European put option price, T is time to maturity and
K is the strike price. Solving for the (unknown) local volatility σ(T, K) gives
a formula for the time inhomogeneous local volatility in terms of derivatives
of the observed European put option prices.
The local volatility model gives the unique martingale diffusion which
is consistent with observed call prices. (Alternative, non-diffusion models
also exist, see for example, Madan and Yor [9].) The recent literature (eg
Schweizer and Wissel [13]) has included attempts to extend the theory to
allow for a stochastic local volatility surface. However, it relies on the knowledge of a double continuum of option prices, which are smooth. In contrast,
Hobson [7] builds models which are consistent with a continuum of strikes,
but at a single maturity, in which case there is no uniqueness.
In the current article we present a method to recover a time-homogeneous
local volatility function from perpetual American option prices. More precisely, we assume that perpetual put option prices are observed for all different values of the strike price, and we derive a time-homogeneous stock
price process for which theoretical option prices coincide with the observed
ones.
No-arbitrage enforces some fundamental convexity and monotonicity conditions on the put prices, and if these fail then no model can support the
observed prices. If the observed put prices are smooth then we can use the
theory of differential equations to determine a diffusion process for which
the theoretical perpetual put prices agree with the observed prices, and our
key contribution in this case is to give an expression for the diffusion coefficient of the underlying model in terms of the put prices. (It turns out
that this expression uniquely determines the volatility co-efficient at price
levels below the current stock price, but the volatility function is undetermined above the current stock price level, except through a single integral
condition.) The key idea is to construct a dual function to the perpetual
put price, and then the diffusion co-efficient can be found easily by taking
derivatives of this dual.
The second contribution of this paper is to give time-homogeneous models
which are consistent with a given set of perpetual put prices, even when
those put prices are not twice differentiable or not strictly convex (in the
continuation region where it is not optimal to exercise immediately). Again
the key is the dual function, coupled with a change of scale and a timechange. We give a construction of a time-homogeneous process consistent
with put prices, which we assume to satisfy the no-arbitrage conditions, but
otherwise has no regularity properties.
One should perhaps note that in reality put prices are only given in the
market for a discrete set of strike prices. Therefore, as a first step one needs
to interpolate between the strikes. If a stock price is modeled as the solution
to a stochastic differential equation with a continuous volatility function,
then the perpetual put price exhibits certain regularity properties with respect to the strike price. Therefore, if one aims at recovering a continuous
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
3
volatility, then one has to use an interpolation method that produces option prices exhibiting this regularity. On the other hand, if a linear spline
method is used, then a continuous volatility cannot be recovered. This is
one of the motivations for searching for price processes which are consistent
with a general perpetual put price function (which is convex but may be
neither strictly convex, nor smooth).
Whilst preparing this manuscript we came across a preprint by Alfonsi and
Jourdain [2]. One of the aims of [2], as in this article, is to construct a timehomogeneous process which is consistent with observed put prices. However
the method is different, and considerably less direct. Alfonsi and Jourdain [2]
build on their previous work [1] to construct a parallel model such that
the put price function in the original model (expressed as a function of
strike) becomes a call price function expressed as a function of the initial
value of the stock. They then solve the perpetual pricing problem for this
parallel model, and subject to solving a differential equation for the optimal
exercise boundaries in this model, give an analytic formula for the volatility
coefficient. In contrast, the approach in this paper is much simpler, and
unlike the method of Alfonsi and Jourdain extends to the irregular case.
2. The Forward Problem
Assume that the stock price process X is modeled under the pricing measure as the solution to the stochastic differential equation
dXt = rXt dt + σ(Xt )Xt dWt ,
X0 = x0 .
Here the interest rate r is a positive constant, the level-dependent volatility
σ : (0, ∞) → (0, ∞) is a given continuous function, and W is a standard
Brownian motion. We assume that the stock pays no dividends, and we let
zero be an absorbing barrier for X. If the current stock price is x0 , then the
price of a perpetual put option with strike price K > 0 is
(1)
P̂ (K) = sup Ex0 e−rτ (K − Xτ )+ ,
τ
where the supremum is taken over random times τ that are stopping times
with respect to the filtration generated by W . From the boundedness, monotonicity and convexity of the payoff we have:
Proposition 2.1. The function P̂ : [0, ∞) → [0, ∞) satisfies
(i) (K − x0 )+ ≤ P̂ (K) ≤ K for all K.
(ii) P̂ is non-decreasing and convex on [0, ∞).
Example. If σ is constant, i.e. if X is a geometric Brownian motion, then
(
K
β if K < K̂
β+1 (βK/x0 (β + 1))
(2)
P̂ (K) =
K − x0
if K ≥ K̂,
where β = 2r/σ 2 and K̂ = x0 (β + 1)/β.
Intimately connected with the solution of the optimal stopping problem
(1) is the ordinary differential equation
1
(3)
σ(x)2 x2 uxx + rxux − ru = 0.
2
4
ERIK EKSTRÖM AND DAVID HOBSON
This equation has two linearly independent positive solutions which are
uniquely determined (up to multiplication with positive constants) if one
requires one of them to be increasing and the other decreasing, see for example Borodin and Salminen [4, p18]. We denote the increasing solution by
ψ̂ and the decreasing one by ϕ̂. In the current setting, ψ̂ and ϕ̂ are given by
ψ̂(x) = x
and
∞
Z y
1
2r
(4)
ϕ̂(x) = x
exp −
dz dy
2
y2
x
c zσ(z)
for some arbitrary constant c. For simplicity, and without loss of generality,
we choose c so that ϕ̂(x0 ) = 1. It is well-known that with Hz = inf{t ≥ 0 :
Xt = z},
ϕ̂(x)/ϕ̂(z) if z < x
x −rHz
(5)
E e
=
ψ̂(x)/ψ̂(z) if z > x.
Z
(This result is easy to check by considering e−rt ϕ̂(Xt ) and e−rt ψ̂(Xt ) which,
since they involve solutions to (3), are local martingales.)
Given the assumed time-homogeneity of the process X, it is natural to
consider stopping times in (1) that are hitting times. Define
P̃ (K) :=
sup Ex0 e−rHz (K − XHz )+
z:z≤x0 ∧K
=
(K − z)Ex0 e−rHz
sup
z:z≤x0 ∧K
=
sup
z:z≤x0 ∧K
K −z
,
ϕ̂(z)
where the last equality follows from (5). By differentiating (4) we find
(6)
Z x0
Z ∞
Z y
2r
1
2r
0
ϕ̂ (x0 ) = exp −
dz
exp −
dz − 1 dy,
2
2
zσ(z)2
c
x0 y
x0 zσ(z)
so ϕ̂0 (x0 ) < 0. Define
K̂ = x0 − 1/ϕ̂0 (x0 ).
Then, for K ≤ K̂ we have
(7)
P̃ (K) = sup
z:z≤K
K −z
K −z
= sup
,
ϕ̂(z)
ϕ̂(z)
z
whereas for K > K̂ we have P̃ (K) = (K − x0 ).
Clearly, P̂ (K) ≥ P̃ (K), and of course, as we show below, there is equality.
Lemma 2.2. The functions P̂ and P̃ coincide, i.e.
K −z
.
(8)
P̂ (K) = sup
z:z≤x0 ϕ̂(z)
Proof. As noted above, clearly P̂ ≥ P̃ since the supremum over all stopping
times is at least as large as the supremum over first hitting times.
For the reverse implication, suppose first that K ≤ K̂. In that case
ϕ̂(z) ≥ (K − z)+ /P̃ (K) by (7). Further, e−rt ϕ̂(Xt ) is a non-negative local
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
5
martingale and hence a supermartingale. Thus, for any stopping time τ we
have
1 ≥ Ex0 [e−rτ ϕ̂(Xτ )] ≥ Ex0 [e−rτ (K − Xτ )+ /P̃ (K)].
Hence P̃ (K) ≥ supτ Ex0 [e−rτ (K − Xτ )+ ] = P̂ (K).
Finally, let K > K̂. It follows from the first part that P̂ (K̂) = K̂ − x0 ,
so Proposition 2.1 implies that P̂ (K) = K − x0 = P̃ (K), which finishes the
proof.
Example. If σ is constant, i.e. if X is a geometric Brownian motion, then
x β
0
,
ϕ̂(x) =
x
where β = 2r/σ 2 . Consequently, the put option price is given by
P̂ (K) = x−β
sup (K − z)z β .
0
z:z≤x0
Straightforward differentiation shows that the supremum is attained for
βK
z = z ∗ :=
β+1
if z ∗ < x0 , and for z = x0 if z ∗ ≥ x0 . Consequently P̂ (K) is given by (2).
Proposition 2.3. There exists a K̂ ∈ (0, ∞) such that P̂ (K) > (K − x0 )+
for all K ∈ (0, K̂) and P̂ (K) = K − x0 for all K ≥ K̂.
Proof. Since σ is continuous and positive, the function ϕ̂(x) given by (4) is
twice continuously differentiable with a strictly positive second derivative.
Therefore, for each fixed K there exists a unique z = z(K) ≤ x0 for which
the supremum in Lemma 2.2 is attained, i.e.
K − z(K)
.
(9)
P̂ (K) =
ϕ̂(z(K))
Geometrically, z = z(K) is the unique value which makes the (negative)
slope of the line through (K, 0) and (z, ϕ̂(z)) as large as possible. Algebraically this means that z = z(K) and K satisfy z(K) = x0 if K ≥ K̂
where
K̂ = x0 − 1/ϕ̂0 (x0 ),
and
(10)
(K − z)ϕ̂0 (z) + ϕ̂(z) = 0
if K < K̂. Therefore, if K ≥ K̂, then
K − x0
= K − x0 ,
P̂ (K) =
ϕ̂(x0 )
and if K < K̂, then
P̂ (K) =
K −z
> K − x0 .
ϕ̂(z)
Proposition 2.4. Suppose σ : (0, ∞) → (0, ∞) is continuous. In addition
to the properties in Propositions 2.1 and 2.3, the following statements about
the function P̂ : [0, ∞) → [0, ∞) hold.
6
ERIK EKSTRÖM AND DAVID HOBSON
(i) P̂ is continuously differentiable on (0, ∞), and twice continuously
differentiable on (0, ∞) \ {K̂}.
(ii) P̂ is strictly increasing on (0, ∞) with a strictly positive second derivative on (0, K̂).
Proof. It follows from (10) and the implicit function theorem that z(K) is
continuously differentiable for K < K̂. Therefore, differentiating (9) gives
(11)
P̂ 0 (K) =
=
(1 − z 0 (K))ϕ̂(z(K)) − (K − z(K))z 0 (K)ϕ̂0 (z(K))
(ϕ̂(z(K)))2
1
,
ϕ̂(z(K))
where the second equality follows from (10). Equation (11) shows that
P̂ 0 (K̂−) = 1/ϕ̂(x0 ) = 1, so P̂ is C 1 also at K̂. Moreover, since ϕ̂(z) is C 1
and z(K) is C 1 away from K̂, it follows that P̂ (K) is C 2 on (0, ∞) \ {K̂}.
In fact, for K < K̂ we have
P̂ 00 (K) =
=
−z 0 (K)ϕ̂0 (z(K))
(ϕ̂(z(K)))2
(ϕ̂0 (z(K)))2
> 0,
(K − z(K))(ϕ̂(z(K)))2 ϕ̂00 (z(K))
where the second equality follows by differentiating (10). Thus P̂ has a
strictly positive second derivative on (0, K̂), which finishes the proof.
Remark. Note that P̂ 0 (0+) ≥ 0 with equality if and only if ϕ̂(0+) = ∞.
We end this section by showing that ϕ̂ can be recovered directly from the
put option prices P̂ (K), at least on the domain (0, x0 ]. To do this, define
the function ϕ : (0, x0 ] → (0, ∞) by
(12)
ϕ(z) = sup
K:K≥z
K −z
P̂ (K)
,
where P̂ is given by (8).
Lemma 2.5.
(i) Suppose f : (0, z0 ] → [1, ∞] is a non-negative, decreasing convex function on (0, z0 ] with f (z0 ) = 1 and f 0 (z0 ) < 0.
Define g : (0, ∞) → [0, ∞) by
(13)
g(k) = sup
z:z≤z0
k−z
.
f (z)
Then g(k) is a non-negative, non-decreasing convex function with
(k − z0 )+ ≤ g(k) ≤ k and g(k) = k − z0 for k ≥ k ∗ = z0 − 1/f 0 (z0 ).
(ii) f and g are self-dual in the sense that if for z ≤ z0 we define
F (z) = sup
k:k≥z
then F ≡ f on (0, z0 ].
k−z
,
g(k)
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
7
(iii) Similarly, assume that g : (0, ∞) → [0, ∞) is a non-negative, nondecreasing convex function with (k − z0 )+ ≤ g(k) ≤ k and g(k) =
k − x0 for k ≥ k ∗ , and define
k−z
f (z) = sup
k:k≥z g(k)
for z ≤ z0 . Then g can be recovered by
k−z
g(k) = sup
.
z:z≤z0 f (z)
Proof. See Appendix A.1.
Corollary 2.6. The function ϕ coincides with the decreasing fundamental
solution ϕ̂ on (0, x0 ].
3. The Inverse Problem: The Regular Case
Now consider the inverse problem. Let P (K) be observed perpetual put
prices for all nonnegative values of the strike K. The idea is that since
ϕ̂ satisfies the Black-Scholes equation (3), Corollary 2.6 provides a way to
recover the volatility σ(x) for x ∈ (0, x0 ] from perpetual put prices. In
this section we provide the details in the case when the observed put prices
are sufficiently regular. We assume that the observed put option price P :
[0, ∞) → [0, ∞) satisfies the following conditions, compare Propositions 2.1,
2.3 and 2.4 above.
Hypothesis 3.1.
(i) (K − x0 )+ ≤ P (K) ≤ K for all K.
(ii) There exists a strike price K ∗ such that P (K) > (K − x0 )+ for all
K < K ∗ and P (K) = K − x0 for all K ≥ K ∗ .
(iii) P is continuously differentiable on (0, ∞), and twice continuously
differentiable on (0, ∞) \ {K ∗ }.
(iv) P is strictly increasing on (0, ∞) with a strictly positive second derivative on (0, K ∗ ).
Motivated by Corollary 2.6, define the function ϕ : (0, x0 ] → (0, ∞) by
K −z
.
(14)
ϕ(z) = sup
K:K≥z P (K)
Proposition 3.2. The function P can be recovered from ϕ by
K −z
P (K) = sup
.
z:z≤z0 ϕ(z)
Proof. This is a consequence of part (iii) of Lemma 2.5.
Proposition 3.3. The function ϕ : (0, x0 ] → (0, ∞) is twice continuously
differentiable with a positive second derivative, and it satisfies ϕ(x0 ) = 1
and ϕ0 (x0 ) = −1/(K ∗ − x0 ).
Proof. For each z ≤ x0 there exists a unique K = K(z) ∈ (z, K ∗ ] for which
the supremum in (14) is attained. Geometrically, K is the unique value
which minimises the slope of the line through (z, 0) and (K, P (K)). Clearly,
K = K(z) satisfies the relation
(15)
(K − z)P 0 (K) = P (K).
8
ERIK EKSTRÖM AND DAVID HOBSON
Reasoning as in the proof of Proposition 2.4 one finds that K(z) is continuously differentiable on (0, x0 ], with
ϕ0 (z) =
(16)
−1
P (K(z))
and ϕ0 (x0 ) = −1/(K ∗ − x0 ). Differentiating (16) with respect to z gives
(17)
ϕ00 (z) =
K 0 (z)P 0 (K(z))
(P 0 (K(z)))2
=
,
P 2 (K(z))
(K(z) − z)P 2 (K(z))P 00 (K(z))
where the second equality follows by differentiating (15). It follows that
ϕ00 (z) is continuous and positive, which finishes the proof.
Next, define σ(x)2 for x < x0 so that ϕ is a solution to the corresponding
Black-Scholes equation, i.e.
(18)
σ(x)2 = 2r
ϕ(x) − xϕ0 (x)
.
x2 ϕ00 (x)
Moreover, define σ(x)2 for x ≥ x0 to be the constant
(19)
σ 2 = 2r(K ∗ − x0 )/x0 .
Now, given this volatility function σ(·), we are in the situation of Section 2,
and can thus define ϕ̂ to be the decreasing fundamental solution to the
corresponding Black-Scholes equation scaled so that ϕ̂(x0 ) = 1. Moreover,
let P̂ (K) be the corresponding perpetual put option price as given by (8).
Theorem 3.4. Assume that Hypothesis 3.1 holds. Then the functions P̂ and
P coincide. Consequently, the volatility σ(x) defined by (18)-(19) solves the
inverse problem.
Proof. First recall that any solution to (3) can be written as a linear combination of ψ̂ and ϕ̂. Consequently, it follows from the definition of σ that
(20)
ϕ(x) = C1 ϕ̂(x) + C2 x
for x ∈ (0, x0 ] and some constants C1 and C2 . The boundary condition
ϕ(x0 ) = 1 yields
(21)
C1 + C2 x0 = 1.
Moreover, from (19) we have
Z y
Z ∞
2r
1
exp −
dz dy
x0
2
2
x0 zσ(z)
x0 y
Z y
Z ∞
1
x0
= x0
exp −
dz dy
2
∗
x0 y
x0 z(K − x0 )
Z ∞
∗
K ∗ /(K ∗ −x0 )
= x0
y −2−x0 /(K −x0 ) dy = (K ∗ − x0 )/K ∗ ,
x0
so the requirement ϕ̂(x0 ) = 1 gives the explicit expression
Z y
Z ∞
K∗
1
2r
ϕ̂(x) = ∗
x
exp
−
dz
dy
2
K − x0 x y 2
x0 zσ(z)
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
9
for the decreasing fundamental solution. Consequently,
Z ∞
Z y
2r
K∗
1
0
ϕ̂ (x0 ) =
exp −
dz dy − 1/x0
2
2
K ∗ − x0
x0 zσ(z)
x0 y
∗
K∗
K − x0
1
=
−
= −1/(K ∗ − x0 ) = ϕ0 (x0 )
K ∗ − x0
K ∗ x0
x0
by Proposition 3.3. It therefore follows from (20) that (1 − C1 )ϕ0 (x0 ) = C2 .
Since ϕ0 (x0 ) < 0, (21) gives that C1 = 1 and C2 = 0, so ϕ = ϕ̂ on (0, x0 ].
Proposition 3.2 then yields
K −z
K −z
P̂ (K) = sup
= sup
= P (K),
z:z≤x0 ϕ̂(z)
z:z≤x0 ϕ(z)
which finishes the proof.
Remark. The inverse problem does not have a unique solution. Indeed, σ
can be defined arbitrarily for x > x0 as long as the condition
Z y
Z ∞
2r
1
exp −
dz dy = (K ∗ − x0 )/(K ∗ x0 )
2
2
x0 zσ(z)
x0 y
is satisfied, which guarantees that ϕ is C 1 at the point x0 .
We next show how to calculate the volatility that solves the inverse problem directly from the observed option prices P (K). To do that, note that
for each fixed z, the supremum in (14) is attained at some K = K(z) for
which
K −z
,
(22)
ϕ(z) =
P (K)
ϕ0 (z) =
(23)
−1
P (K)
and
(24)
ϕ00 (z) =
(P 0 (K))2
,
(K − z)P 2 (K)P 00 (K)
compare (16) and (17). Since ϕ satisfies the Black-Scholes equation, we get
ϕ(z) − zϕ0 (z)
2rKP 2 (K)P 00 (K)
=
.
ϕ00
(P 0 (K))3
Consequently, to solve the inverse problem one first determines z by
P (K)
z=K− 0
,
P (K)
(25)
σ(z)2 z 2 = 2r
and then for this z one determines σ(z) from (25).
4. The inverse problem: the irregular case
Again, suppose we are given perpetual put prices P (K) and an interest
rate r > 0. Our goal is to construct a time-homogeneous process which
is consistent with the given prices. Unlike in the regular case discussed
in Section 3, we now impose no regularity assumptions on the function P ,
beyond the necessary conditions stated in Propositions 2.1 and 2.3. For a
discussion of the necessity of the condition in Proposition 2.3, see Section 9.1.
10
ERIK EKSTRÖM AND DAVID HOBSON
Hypothesis 4.1.
(i) For all K we have (K − x0 )+ ≤ P (K) ≤ K.
(ii) P is non-decreasing and convex.
(iii) There exists K ∗ ∈ (x0 , ∞) such that P (K) = K − x0 for K ≥ K ∗ .
Given P (K) we define K ∈ [0, x0 ] by K = sup{K : P (K) = 0}. Then,
necessarily, P (K) ≡ 0 for K ≤ K.
Theorem 4.2. Given P (K) satisfying Hypothesis 4.1, and given r > 0,
there exists a right-continuous (for t > 0), time-homogeneous Markov process Xt with X0 = x0 such that
sup Ex0 [e−rτ (K − Xτ )+ ] = P (K)
∀K > 0
τ
and such that (e−rt Xt )t≥0 is a local martingale.
Remark. Although we wish to work in the standard framework with rightcontinuous processes, in some circumstances we have to allow for an immediate jump. We do this by making the process right-continuous, except
perhaps at t = 0. At t = 0 we allow a jump subject to the martingale
condition E[X0 ] = x0 .
To exclude a completely degenerate case, henceforth we assume P (x0 ) >
0. If P (x0 ) = 0 then necessarily, to preclude arbitrage, P (K) = (K − x0 )+
and Xt = x0 ert is consistent with the prices P (K). In this case τ ≡ 0 is an
optimal stopping time for every K.
Given P (K) satisfying Hypothesis 4.1 and such that P (x0 ) > 0, define ϕ
by
K −x
(26)
ϕ(x) = sup
K:K≥x P (K)
for x ∈ (0, x0 ]. By Lemma 2.5, ϕ : (0, x0 ] → [1, ∞] is a convex, decreasing,
non-negative function with ϕ(x0 ) = 1. Further,
−
K ∗ − x0 + = ∗
,
K ∗ − x0
K − x0
so ϕ0 (x0 ) ≤ −1/(K ∗ − x0 ) < 0. For some values of x, the supremum in
(26) may be infinite since P may vanish on a non-empty interval (0, K]. We
define
x = inf{x > 0 : ϕ(x) < ∞},
and in the case where x > 0 we see that ϕ(x) = ∞ for x < x. In fact x > 0
if and only if K > 0, and then it is easy to see that these two quantities are
equal.
We extend the definition of ϕ to (x0 , ∞) in any way which is consistent
with the convexity, monotonicity and non-negativity properties and such
0
that limx→∞ ϕ(x) = 0. It is convenient to use ϕ(x) = (x/x0 )ϕ (x0 −)x0 , for
then ϕ0 (x) is continuous at x0 , and ϕ is twice continuously differentiable and
positive on (x0 , ∞).
Given ϕ, define s : (x, ∞) 7→ (−∞, ∞) via s0 (x) = ϕ(x) − xϕ0 (x), with
s(x0 ) = 0. Then s is a concave, increasing function, which is continuous on
(x, ∞). (It will turn out that s is the scale function, which explains the choice
of label.) The function s has a well-defined inverse g : (s(x), s(∞)) → (x, ∞),
and if s(x) > −∞ then we extend the definition of g so that g(y) = x for
ϕ(x0 ) − ϕ(x0 − ) ≤ 1 −
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
11
y ≤ s(x). Note that g : (−∞, s(∞)) → (x, ∞) is a convex, increasing
function with g(0) = x0 . Also define f (y) = ϕ(g(y)). Then f is decreasing
and convex, with f (0) = ϕ(x0 ) = 1.
Example. For geometric Brownian motion, provided β 6= 1 we have
s(x) = xβ0 (x1−β − x1−β
)(1 + β)/(1 − β),
0
y(1 − β) 1/(1−β)
g(y) = x0 1 +
x0 (1 + β)
and
y(1 − β)
f (y) = 1 +
x0 (1 + β)
−β/(1−β)
.
If β = 1 the corresponding formulae are s(x) = 2x0 ln(x/x0 ), g(y) =
x0 ey/(2x0 ) and f (y) = e−y/(2x0 ) .
Remark. Recall that a scale function is only determined up to a linear
transformation. The choice s(x0 ) = 0 is arbitrary but extremely convenient,
as it allows us to start the process Z, defined below, at zero. The choice
s0 (x) = ϕ(x) − xϕ0 (x) is simple, but a case could be made for the alternative
normalisation s0 (x) = (ϕ(x) − xϕ0 (x))/(1 − x0 ϕ0 (x0 )) for which s0 (x0 ) = 1.
Multiplying s by a constant has the effect of modifying the construction
defined in the next section, but only by the introduction of a constant factor
into the time-changes. It is easy to check that this leaves the final model Xt
unchanged.
Our goal is to construct a time-homogeneous process which is consistent
with observed put prices, and such that e−rt Xt is a (local) martingale. In
the regular case we have seen how to construct a diffusion with these properties. Now we have to allow for more general processes, perhaps processes
which jump over intervals, or perhaps processes which have ‘sticky’ points.
One very powerful construction method for time-homogeneous, martingale
diffusions is via a time-change of Brownian motion, and it is this approach
which we exploit.
5. Constructing Time-homogeneous Processes as Time-changes
of Brownian Motion
In this section we extend the construction in Rogers and Williams [12,
Section V.47] of martingale diffusions as time-changes of Brownian motion,
see also Itô and McKean [8, Section 5.1]. The difference from the classical
setting is that the processes defined below may have ‘sticky’ points and may
jump over intervals. Since diffusions by definition are continuous, the resulting processes are not diffusions, but one might think of them as ‘extended
diffusions’, and they are ‘as continuous as possible’.
Let ν be a measure on R and let FB = (FuB )u≥0 be a filtration supporting
a Brownian motion B started at 0 with local time process Lzu . Define Γ to
be the left-continuous increasing additive functional
Z
(27)
Γu =
Lzu ν(dz),
Γ0 = 0
R
12
ERIK EKSTRÖM AND DAVID HOBSON
and let A be the right-continuous inverse to Γ, i.e.
At = inf{u : Γu > t}.
Note that Γ is a non-decreasing process, so that A is well defined, and At
is an FB -stopping time for each time t. Set Z0 = 0 and for t > 0 set
Zt = BAt and Ft = FABt . Note that Z is right-continuous except perhaps at
t = 0. The process Zt is a time-changed Brownian motion adapted to the
filtration F = (Ft )t≥0 and subject to mild non-degeneracy conditions on ν,
(see Lemma 5.1 below), the processes Zt and Zt2 − At are local martingales.
Ru
Rt
Further, if ν(dy) = dy/γ(y)2 then Γu = 0 γ(Br )−2 dr and At = 0 γ(Zs )2 ds,
so that Zt is a weak solution to dZt = γ(Zt )dWt , and Z is a diffusion in
natural scale. The measure ν is called the speed measure of Z, although, as
pointed out by Rogers and Williams, ν is large when Z moves slowly.
The measure ν may have atoms, and it may have intervals on which it
places no mass. If there is an atom at ẑ then dΓu /du > 0 whenever Bu = ẑ,
and then the time-changed process is ‘sticky’ there. Conversely, if ν places
no mass in (α, β) then Γ is constant on any time-periods that B spends in
this interval, and the inverse time-change A has a jump. In particular, Zt
spends no time in this interval. If ν({z̃}) = ∞ then Γu = ∞ for any u
greater than the first hitting time Hz̃B by B of level z̃. Then A∞ ≤ Hz̃B so
that if Z hits z̃, then z̃ is absorbing for Z. The other possibility is that Z
tends to this level without reaching it in finite time.
Define z ν ∈ (0, ∞] and z ν ∈ [−∞, 0) via
z ν = inf{z > 0 : ν((0, z]) = ∞}
and z ν = sup{z < 0 : ν([z, 0)) = ∞}.
The cases where z ν = 0 or z ν = 0 correspond to the degenerate case
Xt = x0 ert mentioned in the previous section, and we exclude them. The
following lemma provides a guide to sufficient conditions for a time-change
of Brownian motion to be a local martingale, and therefore provides insight
into the constructions of local martingales via time-change that we develop
in the next section.
Lemma 5.1. Suppose that either z ν < ∞ or ν charges (a, ∞) for each a,
and further that either z ν > −∞ or ν charges (−∞, a) for each a. Then
Zt = BAt is a local martingale.
Proof. See Appendix A.2.
6. Constructing the model
We now show how to choose the measure ν which gives the process we
want. Define ν via
1 g 00 (y)
(28)
ν(dy) =
dy,
2r g(y)
and let ν(dy) = ∞ for y < s(0) in the case when s(0) > −∞. Similarly,
in the case when s(∞) < ∞ we set ν(dy) = ∞ for y > s(∞). Where g 0 is
absolutely continuous it follows that ν has a density with respect to Lebesgue
measure, but more generally (28) can be interpreted in a distributional sense.
Now, for this ν we can use the construction of the previous section to give
a process Zt . If we then set Xt = g(Zt ), then, subject to the hypotheses of
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
13
Lemma 5.1, Zt = s(Xt ) is a local martingale, so that s is a scale function
for X. The process X is our candidate process for which the associated put
prices are given by P .
Example. For geometric Brownian motion,
ν(dy)
β
y(1 − β) −2
1 g 00 (y)
1
1+
.
=
=
dy
2r g(y)
x0 (1 + β)
2rx20 (1 + β)2
In the case β = 1 this simplifies to
ν(dy)
1
1
= 2 = 2
dy
C
8x0 r
√
D
tC 2 ; Zt = BtC 2 = C B̃t for a
where C = 2x0 2r. Then Γu = uC −2 ; At = √
Brownian motion B̃ and Xt = x0 eZt /2x0 = x0 e 2rB̃t .
R
Recall that Γu = R Lzu ν(dz) and let ξ be the first explosion time of Γ.
Note that by construction Γ is continuous for t < ξ, and left continuous at
t = ξ. Since ν is infinite outside the interval [s(0), s(∞)] we also have the
B ∧ HB
expression ξ = inf{u : Bu ∈
/ [s(0), s(∞)]} = Hs(0)
s(∞) . The inverse scale
function g is concave on (s(0), s(∞)), but may have a jump (from a finite to
an infinite value) at s(∞). In that case we take it left-continuous at s(∞)
so that we may have ḡ := limz↑∞ g(s(z)) is finite.
For 0 < u < ξ, define Mu = e−rΓu g(Bu ) and Nu = e−rΓu f (Bu ).
Lemma 6.1. M = (Mu )0≤u<ξ and N = (Nu )0≤u<ξ are FB -local martingales.
Sketch of proof: Suppose that ϕ is twice continuously differentiable with
a positive second derivative. Then g is twice continuously differentiable. For
u < ξ, applying Itô’s formula to Mu = e−rΓu g(Bu ) gives
dΓu
1
erΓu dMu = g 0 (Bu )dBu + −r
g(Bu ) + g 00 (Bu ) du.
du
2
But, by definition, dΓu /du = g 00 (Bu )/(2rg(Bu )), so M is a local martingale
as required.
A similar argument can be provided for the process N . For the general
case, see the Appendix.
Since M and N are non-negative local martingales on [0, ξ) they converge
almost surely to finite values, which we label Mξ and Nξ . In particular,
B , then M = 0. However, if ξ = H B
if ξ = Hs(0)
ξ
s(∞) then there are several
cases. The fact that a non-negative local martingale converges means that
we cannot have both Γξ < ∞ and ḡ = limz↑∞ g(s(z)) = ∞. Instead, if
Γξ < ∞ then ḡ < ∞ and Mξ = e−rΓξ ḡ. If Γξ = ∞, and ḡ < ∞ then Mξ = 0,
whereas if Γξ = ∞ and ḡ = ∞ then (Mu )u<ξ typically has a non-trivial
limit. Similar considerations apply to N .
Recall that A is the right-continuous inverse to Γ and define the timechanged processes M̃t = MAt and Ñt = NAt . Note that these processes
are adapted to F and that, at least for t ≤ Γξ , we have ΓAt = t, M̃t =
e−rt g(Zt ) = e−rt Xt and Ñt = e−rt f (Zt ) = e−rt ϕ(Xt ).
If (s(0), s(∞)) = R then ξ = ∞, Γξ = ∞ and M̃ is defined for all t.
14
ERIK EKSTRÖM AND DAVID HOBSON
B . In this case either Γ =
If s(0) > −∞ then we may have ξ = Hs(0)
ξ
ΓH B
s(0)
= ∞, whence M̃ is defined for all t as before, or Γξ < ∞. Then
M̃Γξ = Mξ = e−rΓξ g(Bξ ) = 0, and we set M̃t = 0 for all t > Γξ . It follows
that Xt = 0 for all t ≥ Γξ and 0 is an absorbing state.
B
Similarly, if s(∞) < ∞ then we may have ξ = Hs(∞)
. Then either Γξ = ∞,
B
whence M̃ is defined for all t, or Γξ < ∞. In the latter case, if ξ = Hs(∞)
<
∞ and Γξ < ∞ then M̃Γξ = Mξ = e−rΓξ ḡ. We set M̃t = M̃Γξ for all t > Γξ ,
and it follows that for t > Γξ , Xt := ert M̃t = er(t−Γξ ) ḡ. Thus, for t > Γξ , X
grows deterministically. An example of this situation is given in Example 8.4
below. (In fact, the case where ḡ < ∞, which depends on the behaviour of
the scale function s to the right of x0 , can always be avoided by suitable
choice of the extension to ϕ.)
We want to show how M̃ and X inherit properties from M . The key
idea below is that, loosely speaking, a time-change of a martingale is again
a martingale. Of course, to make this statement precise we need strong
control on the time-change. (Without such control the resulting process can
have arbitrary drift. Indeed, as Monroe [10] has shown, any semi-martingale
can be constructed from Brownian motion via a time-change.) We have the
following result the proof of which is given in the Appendix.
Corollary 6.2. The process (e−rt Xt )t≥0 is a local martingale.
We can perform a similar analysis on N and Ñ and use similar ideas to
ensure that Ñ is defined on R+ . The proof that Ñ is a local martingale
mirrors that of Corollary 6.2.
Corollary 6.3. The process (e−rt ϕ(Xt ))t≥0 is a local martingale.
7. Determining the put prices for the candidate process
Recall the definitions of s, g and ν via s0 (x) = ϕ(x) − xϕ0 (x), g ≡ s−1
and ν(dy) = g 00 (y)/(2rg(y))dy. Suppose that Z is constructed from ν and
a Brownian motion using the time-change Γ, and construct the candidate
price process via Xt = g(Zt ). By Corollary 6.2, the discounted price e−rt Xt
is a (local) martingale. To complete the proof of Theorem 4.2 we need to
show that for the candidate process Xt the function
P̂ (K) := sup E[e−rτ (K − Xτ )+ ]
τ
is such that P̂ (K) ≡ P (K) for all K ≥ 0.
Unlike the regular case the process X that we have constructed may have
jumps. For this reason, for x < x0 we modify the definition of the first
hitting time so that Hx = inf{u > 0 : Xu ≤ x}.
Theorem 7.1. The perpetual put prices for X are given by P .
Proof: Fix x ∈ (x, x0 ). Suppose first that x is such that Γ is strictly
increasing whenever the Brownian motion B takes the value s(x). Then
XHx = x. More generally the same is true whenever ν((s(x) − δ, s(x)]) > 0
for every δ > 0. By Corollary 6.3 we have that (e−rt ϕ(Xt ))t≤Hx is a local
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
15
martingale, and ϕ is bounded on [x, ∞) so it follows that e−r(t∧Hx ) ϕ(Xt∧Hx )
is a bounded martingale and ϕ(x0 ) = Ex0 [e−rHx ϕ(x)]. Hence,
P̂ (K) ≥ Ex0 [e−rHx (K − x)] = (K − x)
ϕ(x0 )
K −x
=
.
ϕ(x)
ϕ(x)
Otherwise, fix x− (x) = inf{w < x : ν((s(w), s(x)]) = 0} and x+ (x) =
sup{w > x : ν([s(x), s(w)) = 0}. It must be the case that ϕ is linear on
(x− (x), x+ (x)) and bounded on [x− (x), ∞) and
P̂ (K) ≥
=
max
w∈{x− ,x+ }
Ex0 [e−rHw (K − w)]
K −w
K −x
≥
.
ϕ(x)
w∈{x− ,x+ } ϕ(w)
max
It follows that
(29)
P̂ (K) ≥ sup
x:x≤x0
K −x
= P (K).
ϕ(x)
(Clearly, if x < x then (K − x)/ϕ(x) = 0, so the supremum cannot be
attained for such an x.)
To prove the reverse inequality, we first claim that the left derivative D− ϕ
of the convex function ϕ satisfies
(30)
D− ϕ(x0 ) := lim
↓0
−1
ϕ(x0 ) − ϕ(x0 − )
= ∗
.
K − x0
To prove (30), first note that choosing K = K ∗ in (26) yields ϕ(x) ≥
(K ∗ − x)/(K ∗ − x0 ), so
D− ϕ(x0 ) = lim
x↑x0
−1
ϕ(x0 ) − ϕ(x)
≤ ∗
.
x0 − x
K − x0
Conversely, note that for each δ > 0 there exists a non-empty interval (x0 −
, x0 ) on which
K∗ − δ − x
ϕ(x) ≤ ∗
.
K − δ − x0
Consequently, for x ∈ (x0 − , x0 ) we have
ϕ(x0 ) − ϕ(x)
−1
≥ ∗
.
x0 − x
K − δ − x0
Thus D− ϕ(x0 ) ≥ −1/(K ∗ − x0 ) since δ > 0 is arbritrary, so (30) follows.
We next claim that for each fixed K ≤ K ∗ we have
(31)
ϕ(x) ≥ (K − x)+ /P (K)
for all x. Clearly, this holds for x ≥ K and for x ≤ x0 . Similarly, if
x0 < x < K, then it follows from (30) and the convexity of ϕ that
ϕ(x) ≥
K∗ − x
K −x
K −x
≥
≥
.
∗
K − x0
K − x0
P (K)
It follows from (31) and Corollary 6.3 that for any stopping rule τ we have
Ex0 [e−rτ (K − Xτ )+ ] ≤ P (K)Ex0 [e−rτ ϕ(Xτ )] ≤ P (K)ϕ(x0 ) = P (K).
Hence P̂ (K) ≤ P (K) for K ≤ K ∗ , and in view of (29), P̂ (K) = P (K).
16
ERIK EKSTRÖM AND DAVID HOBSON
For K > K ∗ it follows from P̂ (K ∗ ) = P (K ∗ ) = K ∗ − x0 , the convexity
of P̂ and Hypothesis 4.1 that P̂ (K) = K − x0 = P (K), which finishes the
proof.
8. Examples
The following examples illustrate the construction of the previous sections. The list of examples is not intended to be exhaustive, but rather
indicative of the types of behaviour that can arise. In each example we
assume x0 = 1.
8.1. The smooth case. We have studied the case of exponential Brownian
motion throughout. It is very easy to generate other examples, for example by choosing a smooth decreasing convex function (with ϕ(x0 ) = 1 and
limx↑∞ ϕ(x) = 0), and defining other quantities from ϕ.
Example 8.1. Suppose ϕ(x) = (x + 1)/(2x2 ). Then, from (3) we obtain
σ 2 (x) = r
2x + 3
x+3
x>0
and, from (8)
(K + 9)3/2 (K + 1)1/2 − (27 + 18K − K 2 )
4
with P (K) = (K − 1) for K ≥ 5/3.
P (K) =
K ≤ 5/3
8.2. Kinks in P . If the first derivative of P is not continuous then we find
that ϕ is linear over an interval (α, β) say. Then s0 is constant on this
interval and g is linear over the interval (s(α), s(β)). It follows that ν does
not charge this interval, so that Γu is constant whenever Bu ∈ (s(α), s(β)),
and At has a jump. Then Zt jumps over the interval (s(α), s(β)), and Xt
spends no time in (α, β).
Example 8.2. Suppose P (K) satisfying Hypothesis 4.1 is given by
 2
0 < K ≤ 27/32
 K /8
3
4K /27
27/32 ≤ K ≤ 3/2
P (K) =

(K − 1)
3/2 ≤ K.
Then P is continuous, but P 0 has a jump at K = 27/32.
Using (26) we find that
2x−1 0 < x ≤ 27/64
ϕ(x) =
x−2 x > 9/16.
Over the region I = [27/64, 9/16] ϕ is given by linear interpolation. The
corresponding scale function is linear on I, and in the construction of Z,
ν assigns no mass to s(I). The process
with
√ X is a generalised diffusion
√
diffusion coefficient given by σ(x) = 2r for x ≤ 27/64, σ(x) = r for
x ≥ 9/16 (strictly speaking there is√some freedom in the choice of σ for
x ≥ x0 ≡ 1, but the constant value r is a natural choice) and σ(x) = ∞
for x ∈ I.
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
17
8.3. Linear parts to P . In this case the derivative of ϕ(x) is discontinuous
at a point γ say. Then s0 is also discontinuous at this point, and g 0 is
discontinuous at s(γ). It follows that ν has a point mass at s(γ), and that
Γu includes a multiple of the local time at s(γ).
Example 8.3. Suppose P (K) satisfying Hypothesis 4.1 is given by

8K 3 /27
0 < K ≤ 3/4



(2K − 1)/4
3/4 ≤ K ≤ 1
P (K) =
K 2 /4
1≤K≤2



(K − 1)
2 ≤ K.
Then P is convex, but is linear on the interval [3/4, 1].
Then
−2
x /2 0 < x ≤ 1/2
ϕ(x) =
x−1
x > 1/2
where we have chosen to extend the definition of ϕ to (1, ∞) in the natural
way. Then s(x) = 3 − 2 ln 2 − 3/2x for x < 1/2 and s(x) = 2 ln x otherwise.
It follows that g is everywhere convex but has a discontinuous first derivative
at z = −2 ln 2, and that the corresponding measure ν has a positive density
with respect to Lebesgue measure and an atom of size r−1 /12 at −2 ln 2.
In the terminology of stochastic processes the process Z is ‘sticky’ at this
point — for a discussion of sticky Brownian motion see Amit [3] or, for the
one-sided case Warren [14].
If P is piecewise linear (for example if P is obtained by linear interpolation from a finite number of options) then ϕ is piecewise linear, s is piecewise
linear, g is piecewise linear, and ν consists of a series of atoms. As a consequence the process Zt is a continuous-time Markov process on a countable
state-space (at least whilst Zt < s(x0 ) ≡ 0), in which transitions are to
nearest neighbours only. Holding times in states are exponential and the
jump probabilities are such that Zt is a martingale.
In turn this means that Xt is a continuous-time Markov process on a
countable set of points (at least whilst Xt < x0 ).
Example 8.4. Suppose

K≤1
 K/3
(2K − 1)/3 1 ≤ K ≤ 2
P (K) =

(K − 1)
K ≥ 2.
This is consistent with a situation in which only two perpetual American
put options trade, with strikes 1 and 3/2 and prices 1/3 and 2/3, in which
case we may assume that we have extrapolated from the traded prices to a
put pricing function P (K) which is consistent with the traded prices.
Consider the process Xt with state space {0} ∪ {1/2} ∪ [2, ∞) and such
that
• at t = 0+, X jumps to 1/2 or 2 with probabilities 2/3 and 1/3, respectively;
• if ever Xt0 ≥ 2 then thereafter Xt = Xt0 er(t−t0 ) ;
• zero is an absorbing state for X;
• if ever X reaches 1/2 then it stays there for an exponential length of time,
rate 3r, and jumps to 2 with probability 1/3 and zero with probability 2/3.
18
ERIK EKSTRÖM AND DAVID HOBSON
Note that for the continuous time Markov process Xt , then conditional
on Xt = 1/2,
1
1
2
r
lim E[Xt+∆ − Xt ] ∼ 3r (2 − Xt ) + (−Xt ) = = rXt
∆↓0 ∆
3
3
2
Also, for this process
P (K) =
max
τ =0,H1/2 ,H0
E[e−rτ (K − Xτ )+ ]
= max{K − 1, (2K − 1)/3, K/3}
so we recover the put price function given at the start of the example.
8.4. Positive gradient of P at zero, i.e. P 0 (0) > 0. In this case
limx↓0 ϕ(x) < ∞. It follows that s(0) > −∞ and the resulting diffusion
Xt can hit zero in finite time. In this case we insist that 0 is an absorbing
endpoint for the diffusion.
Example 8.5. Suppose that

K<1
 K/2
(K + 1)2 /8 1 ≤ K ≤ 3
P (K) =

K −1
K ≥ 3.
−1
2
2
Then ϕ(x)
p = 2(x + 1) and x σ(x) = r(x + 1)(2x + 1), so that dXt =
rXt + r(Xt + 1)(2Xt + 1) dBt .
The following example covers the case of mixed linear and smooth parts
to P (K), and shows an example where reflection, local times and jumps all
form part of the construction.
Example 8.6. Suppose P (K) satisfies

K≤1
 K/4
K 2 /4
1≤K≤2
P =

(K − 1) K ≥ 2.
Note that P 0 has a jump at x = 1/2. We have ϕ(x) = 4 − 4x for x < 1/2
and ϕ(x) = 1/x for 1/2 ≤ x ≤ 1. We assume this formula applies on [1, ∞)
also.
It follows that for x ≥ 1/2, η(x)2 ≡ (xσ(x))2 = 2rx2 . Note that since
ϕ(1/2) < ∞ we have H1/2 is finite (almost surely). Hence
√
t ≤ H1/2
(32)
dXt = rXt dt + 2rXt dBt
is consistent with the observed put prices, but we need to describe
what
√
happens when X hits 1/2. Note that (32) has solution Xt = e 2rBt for
t ≤ H1/2 .
The process is not absorbed at 1/2 as this would conflict with the idea
that the discounted price process is a local martingale, or equivalently dXt =
rXt dt + dMt , for a local martingale Mt . However, adding a reflection alone
would again destroy the martingale property of e−rt Xt . What the general
construction above does is that a local time reflection and a compensating
downward jump are added. This jump takes the process to zero where it is
absorbed.
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
19
Alternatively,
the process can be formalised as follows. Let It = − inf u≤t {(Bu +
√
ln 2/ 2r)∧0}. Then by Skorokhod’s Lemma, Bt +I
is a reflected Brownian
√ t
√
motion (reflected at the level −(ln 2/ 2r)) and e 2r(Bt +It ) ≥ 1/2.
Let N λ be a Poisson Process rate λ, independent of B, and let T λ be the
first event time. Then the compensated Poisson Process (Ntλ − λt)t≥0 and
λ
the compensated Poisson Process stopped at the first jump (Nt∧T
λ − λ(t ∧
λ
λ
λ
T ))t≥0 are martingales. The time change (NIt ∧T λ − λ(It ∧ T ))t≥0 , is also
a martingale.√
Take λ = 2r and define X via X0 = 1 and
√ !
√
√
dNIt 2r
dXt = rXt dt + 2rXt dBt + dIt − √
t : It ≤ T 2r
2r
Note that at the first jump time of the time-changed Poisson process, X
jumps from 1/2 to zero.
By construction (e−rt Xt )t≥0 is a martingale.
8.5. P is zero on an interval: P (K) = 0 for K ≤ K. Now we find
that ϕ(x) = ∞ for x ≤ x where x = K. Depending on whether the right
derivative P 0 (K+) is zero or positive, we may have ϕ(x+) is infinite or finite.
In the former case we have that Xt does not reach x in finite time. In the
latter case Xt does hit x in finite time.
The first example is typical of the case where ϕ(x+) = ∞, or equivalently
where there is smooth fit of P at K.
Example 8.7. Suppose X0 = 1 and that

 0
(2K − 1)2 /8
P (K) =

K −1
P (K) solves
K ≤ 1/2
1/2 ≤ K ≤ 3/2
K ≥ 3/2.
Then P 0 is continuous and for 1/2 < x < 1 we have ϕ(x) = (2x − 1)−1 . We
also have ϕ(x) = ∞ for x ≤ 1/2. It does not matter how ϕ is continued on
(1, ∞) (provided it is decreasing and convex), but for definiteness we assume
the formula ϕ(x) = (2x − 1)−1 applies there also.
It follows that η(x)2 ≡ (xσ(x))2 = r(2x − 1)(4x − 1)/4. Note that since
ϕ(1/2) = ∞ we have H1/2 (the first hitting time of 1/2) is infinite. Hence
r
r(2x − 1)(4x − 1)
dBt
t ≤ H1/2
dXt = rXt dt +
4
is consistent with the observed put prices, and since the process never hits
1/2, it is not necessary to describe the process beyond H1/2 .
Now consider the other case where P 0 (K) > 0.
Example 8.8. Suppose X0 = 1 and that

0



(2K − 1)/4
P (K) =
K 2 /4



K −1
Then P 0 has a jump at K = 1/2.
P (K) solves
K ≤ 1/2
1/2 ≤ K ≤ 1
1≤K≤2
K ≥ 2.
20
ERIK EKSTRÖM AND DAVID HOBSON
We have ϕ(x) = ∞ for x < 1/2 and ϕ(x) = 1/x for 1/2 ≤ x ≤ 1. We
assume this formula applies on [1, ∞) also.
It follows that for x ≥ 1/2, η(x)2 ≡ (xσ(x))2 = 2rx2 . As before we have
√
(33)
dXt = rXt dt + 2rXt dBt
t ≤ H1/2
is consistent with the observed put prices, but we need to describe what
happens when X hits 1/2.
The probability that the process ever goes below 1/2 is zero, else the
put with strike K = 1/2 would have positive value. The process cannot
be absorbed at 1/2 as this would conflict with the idea that the discounted
price process is a martingale, or equivalently dXt = rXt dt + dMt , for a local
martingale Mt . Instead, the time-change Γ includes a multiple of the local
time at s(1/2) = −2 ln 2.
In fact g 00 /g is constant for y > s(1/2) and equal to zero for y < s(1/2).
ln 2 , where O + is the
The process Γu is a linear combination of Ou+ and L−2
u
u
amount of time spent by the Brownian motion above s(1/2) before time u.
It is easy to check using Itô’s formula that e−rΓu g(Bu ) is a martingale in
this case. The process Zt = BAt is ‘sticky’ at s(1/2) (this time in the sense
of a one-sided sticky Brownian motion, see Warren [14]) and this property
is inherited by X = s(Z).
There is a third case, where ϕ(x+) < ∞, but ϕ0 (x+) = ∞.
√
Example 8.9. Suppose ϕ(x) = 2 − 2x − 1 for 1/2 ≤ x ≤ 5/2. Equivalently,

K ≤ 1/2
 0 √
P (K) =
2 − 5 − 2K 1/2 ≤ K ≤ 2

K −1
K ≥ 2.
Then for 1/2 < x < 5/2 we have
√
η(x)2 = (xσ(x))2 = 2r(2x − 1) 2 2x − 1 + 1 − x
It follows that although Xt can hit 1/2, the volatility at this level is zero,
and the drift alone is sufficient to keep Xt ≥ 1/2.
8.6. Kink in P at K ∗ : K ∗ < ∞ and P 0 (K ∗ −) < 1. In this case ϕ0 (x) is
constant on an interval (x̂, x0 ). This case is just like Section 8.2.
9. Extensions
9.1. No options exercised immediately. In Hypothesis 4.1, in addition
to (i) and (ii) which are enforcible by no-arbitrage considerations, we also
assumed (iii) that there exists a finite strike K ∗ such that for all strikes K ≥
K ∗ the put option is exercised immediately. Since K ∗ < ∞ is equivalent
to ϕ0 (x0 ) < 0, it is apparent from the expression in (6), that provided σ is
finite for some x > x0 , or equivalently ν gives mass to some z > 0, then
this property will hold. However, it is interesting to consider what happens
when this fails.
Suppose P (K) > K − x0 for all K and limK P (K) − (K − x0 ) = 0. Then
0
ϕ (x0 ) = 0, but ϕ is strictly decreasing on (x, x0 ). The measure ν places
no mass on (0, ∞), the process Zt spends no time on (0, ∞) and Xt never
takes values above x0 . In particular Xt is reflected (downwards) at x0 . The
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
21
resulting model is consistent with observed option prices, but not with the
assumption that the discounted price process is a martingale. However,
by allowing non-zero dividend rates we can find a model for which the exdividend price process is a martingale, and for which the model prices are
given by P (K). See Section 9.2 below.
Now suppose limK P (K) − (K − x0 ) = δ > 0. If P (x0 ) = x0 then
P (K) ≡ K, and we have an extreme example which falls into this setting.
For P as specified above we have that ϕ(x) = 1 on (x0 − δ, ∞). The measure
ν places no mass on (−δ, ∞) and s(x) = x − x0 on this region. Except for
time 0, the process Zt spends no time in (−δ, ∞) and Xt jumps instantly to
x0 − δ, and thereafter spends no time above this point.
Note that if x0 is not specified, then this case can be reduced to the
previous case by assuming x0 = K − limK P (K). This obviates the need for
a jump at t = 0.
9.2. Time homogeneous processes with non-constant interest rates
and non-zero dividend processes. In the main body of the paper we
have assumed that the interest rate r is a given positive constant, that dividend rates are zero, and that σ is a function to be determined. However, the
same ideas can be used to find other time-homogeneous models consistent
with observed perpetual put prices, whereby the volatility function is given,
and either a state-dependent dividend rate, or a state-dependent interest
rate is inferred.
Suppose X has dynamics dXt = (r(Xt ) − q(Xt ))Xt dt + Xt σ(Xt ) dBt .
Given put prices P (K) as before, define ϕ via ϕ(z) = inf K:K≥z (K−z)/P (K).
Then the relationship between ϕ and the characteristics of the price process
X are such that ϕ solves Lϕ = 0, where L is given by
1
Lu = σ(x)2 x2 uxx + x(r(x) − q(x))ux − r(x)u.
2
Note that we now allow for any of σ, q or r to depend on x. To date we have
assumed that r is constant and q is zero and solved for σ, but alternatively
we can assume σ(x) is a given function, and r is a positive constant, and
solve for q, or assume that q and σ are given, and solve for r.
For example, if r and σ are given constants then the (proportional) dividend rate process is given by
x2 σ 2 ϕxx − 2rϕ
.
2xϕx
If q is negative this should be thought of as a convenience yield.
By allowing for dividend processes which are singular with respect to
calender time, and which are instead related to the local time of X at level
x0 , it is possible to construct candidate price processes which spend no time
above x0 . For example, if L̃ is the local time at 1 of X, and if
q(x) = r +
dXt
dL̃t
= dBt + rdt −
Xt
2
then Xt reflects at 1, and if ϕ̂(x) = (E1 [e−rHx ])−1 for x < 1, then ϕ̂0 (1−) = 0.
This gives an example of a model consistent with the class of option prices
described in Section 9.1.
22
ERIK EKSTRÖM AND DAVID HOBSON
9.3. Recovering the model from perpetual calls. The perpetual American call price function C : [0, ∞) → [0, x0 ] must be non-increasing and convex as a function of the strike K, and must satisfy the no-arbitrage bounds
(x0 − K)+ ≤ C(K) ≤ x0 .
If there are no dividends (and if e−rt Xt is a martingale) then the perpetual
call prices are given by the trivial function C(K) = x0 .
So suppose instead that the (proportional) dividend rate q is positive. Let
ψ̂ be the increasing positive solution to
1 2
x σ(x)2 ψ 00 + (r − q)xψ 0 − rψ = 0
2
normalised so that ψ̂(x0 ) = 1. Then, for z > x, Ex [e−rHz ] = ψ̂(x)/ψ̂(z), and
call prices in a model where dXt = (r − q)Xt dt + Xt σ(Xt ) dBt are given by
Ĉ(K) = sup Ex0 [e−rτ (Xτ − K)+ ] = sup
τ
x:x≥x0
(x − K)
ψ̂(x)
.
Example. Suppose X solves (dXt /Xt ) = (r − q) dt + σ dBt , with X0 = x0 .
Then ψ̂(x) = (x/x0 )γ where γ = β+ and
s
r−q 1
2r
r−q 1 2
β± = −
−
±
+
−
.
σ2
2
σ2
σ2
2
Note that since q > 0 we have γ > 1. Note also that ϕ̂(x) = xβ− .
The corresponding call prices are given by
Ĉ(K) = xγ0 sup {(x − K)x−γ }
x:x≥x0
which for K ≤ (γ − 1)x0 /γ gives Ĉ(K) = (x0 − K), and for K > (γ − 1)x0 /γ
Ĉ(K) = xγ0 γ −γ (γ − 1)γ−1 K 1−γ .
The example discusses the forward problem, but the discussion of the
inverse problem is similar to that in the put case. Given perpetual call prices
C(K), for x > x0 we can define ψ via ψ(x) = inf K:K≤x (x − K)/C(K), and
then construct a triple σ(x), q(x), r(x) so that
1 2
x σ(x)2 ψ 00 + (xr(x) − q(x))ψ 0 − r(x)ψ = 0.
2
By combining information from put and call prices it is possible to determine
a model which simultaneously matches both puts and calls.
Appendix A. Proofs
A.1. Duality.
Proof of Lemma 2.5. It is clear that g is non-negative and non-decreasing
since f is positive and non-increasing. The lower bound on g follows from
choosing z = z0 ∧ k in (13), and the upper bound follows since f is nonincreasing. To show that g is convex, first note that g(k) is minus the
reciprocal of the slope of the tangent of the function f which passes through
the point (k, 0). For two given points k1 and k2 with k1 < k2 , let l1 (z)
and l2 (z) be the corresponding tangent lines. Let k = λk1 + (1 − λ)k2 for
some λ ∈ (0, 1), and let l(z) be the line through the point (0, k) and the
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
23
intersection point of l1 and l2 . If the intersection point is denoted (z, l(z)),
then the convexity of f guarantees that
g(k) ≤
k−z
(1 − λ)k1 − (1 − λ)z λk2 − λz
=
+
= (1 − λ)g(k1 ) + λg(k2 ),
l(z)
l1 (z)
l2 (z)
which proves that g is convex.
To prove the self-duality, let z ≤ z0 . By the definition of g we have that
g(k) ≥ (k − z)/f (z) for all k ≥ z. Consequently,
F (z) = sup
k≥z
k−z
≤ f (z).
g(k)
For the reverse inequality, let z ≤ z0 and let l be a tangent line to f through
the point (z, f (z)) (such a tangent is not necessarily unique if f has a kink
at z). Assume that the point where l intersects the z-axis is given by (k 0 , 0).
Then g(k 0 ) = (k 0 − z)/f (z), so
F (z) = sup
k≥z
k0 − z
k−z
≥
= f (z),
g(k)
g(k 0 )
which finishes the proof of (ii). The proof of (iii) can be performed along
the same lines.
A.2. Time changes of local martingales.
Proposition A.1. Suppose (γu )u≥0 is a martingale with respect to the filtration G = (Gu )u≥0 , and At is an increasing process such that At is a stopping
time with respect to G for each t. Define γ̃t = γAt , and G̃t = GAt . In general
(γ˜t )t≥0 is not a martingale. However, if γ is a bounded martingale then γ̃
is a bounded martingale.
Proof. Given a Brownian motion B, for b > 0, let HbB be the first hitting
time of level b. Then (B̃b )b≥0 defined via B̃b ≡ BH B is not a martingale.
b
However, if γ is bounded then E[γ̃t |G̃s ] = E[γAt |GAs ] = γAs = γ̃s by
optional sampling.
Suppose now that we are in the setting of Section 5 where Zt is constructed
from the Brownian motion B. In particular, Γu is an increasing additive
functional of B, and A is the right-continuous inverse to Γ.
Proof of Lemma 5.1. Intuitively a time change of Brownian motion is a
local martingale, but if the additive functional Γ is constant when B is in
[a, ∞) then the resulting process spends no time above a and reflects there.
To maintain the local martingale property we need either that the timechanged process never gets to a, or that there are arbitrarily large values at
which Γ is strictly increasing.
If [z ν , z ν ] is a bounded interval, then A∞ ≤ HzBν ∧ HzBν and (Zt )0≤t<∞ =
(BAt )0≤At <A∞ is a bounded martingale by Proposition A.1.
Now suppose (z ν , z ν ) = R and suppose that for each a, ν assigns mass to
every set (a, ∞) and (−∞, a).
We have AΓt ≥ t with equality provided Γ is strictly increasing at t. Let
−
{a+
n } and {an } be two sequences converging to +∞ and −∞, respectively,
−
so that ν assigns mass to any neighborhood of a+
n and an , and set Hn =
24
ERIK EKSTRÖM AND DAVID HOBSON
+
inf{u : Bu ∈
/ (a−
n , an )}. Then Γ is strictly increasing at Hn . Set Tn = ΓHn .
Then ATn = Hn . Note that Γu increases to infinity almost surely, and hence
ΓHn ↑ ∞. Under our hypothesis, (MtTn )t≥0 given by
MtTn := Mt∧Tn = BAt∧Tn = BAt ∧Hn
is a bounded martingale. Hence Tn is a localisation sequence for M .
The mixed case can be treated similarly.
Proof of Lemma 6.1. For y ∈ (s(0), s(∞)), set H(y) = ln(g(y)/x0 ) and
write h(y) = H 0 (y) = g 0 (y)/g(y). Then
1 g 00 (y)
1
ν(dy) =
dy =
h0 (dy) + h(y)2 dy
2r g(y)
2r
and, as usual, ν(dy)R= ∞ for y ∈
/ [s(0), s(∞)].
y
We have H(y) = 0 h(v)dv = ln(g(y)/x0 ) and
Z
1
Γu =
Ly (H 00 (dy) + H 0 (y)2 dy).
2r R u
Let ξ be the first explosion time of Γ. Then, by the Itô-Tanaka formula (eg,
Revuz and Yor [11, Theorem VI.1.5]), for u < ξ,
Z u
Z
1
0
H(Bu ) =
H (Bs )dBs +
H 00 (y)Lyu dy
2 R
0
Z u
Z
1
=
h(Bs )dBs −
Lyu h(y)2 dy + rΓu .
2
0
R
Thus g(Bu ) = x0 eH(Bu ) = E(h(B) · B)u erΓu , where E denotes the Doléans
exponential, and e−rΓu g(Bu ) is a local martingale. It follows that M is a
local martingale.
Ry
Now define J(y) = 0 j(v)dv = ln f (y) and
Z
1
Γ̃u =
Lyu J 00 (dy) + J 0 (y)2 dy
2r R
By exactly the same argument as above we find that f (Bu ) = eJ(Bu ) =
E(j(B) · B)u erΓ̃u and e−rΓ̃u f (Bu ) is a local martingale.
It remains to show that in fact Γu = Γ̃u . Define L(y) = (f (y)g(y))−1 .
Then L0 (y)/L(y) = −g 0 (y)/g(y) − f 0 (y)/f (y) = −(H 0 (y) + J 0 (y)) and
ϕ0 (g(y))g 0 (y) g 0 (y)
g 0 (y)[g(y)ϕ0 (g(y)) − ϕ(g(y))]
−
=
ϕ(g(y))
g(y)
g(y)ϕ(g(y))
0
0
g (y)s (g(y))
= −
= −L(y)
g(y)f (y)
J 0 (y) − H 0 (y) =
and then
(J 0 (y) − H 0 (y))0 = (H 0 (y) + J 0 (y))(L(y)) = H 0 (y)2 − J 0 (y)2 .
Finally, since Lyu is a bounded continuous function with compact support
for each fixed u, we conclude that Γu = Γ̃u .
Proof of Corollary 6.2. Recall that in our setting Γ defined via (27)
grows without bound and is continuous, at least until B hits s(0) or s(∞).
Thus, if ξ denotes the first explosion time of Γ, then the inverse function A
RECOVERING A TIME-HOMOGENEOUS STOCK PRICE PROCESS
25
is defined for every t and At = ξ for t ≥ Γξ . Then, using the extension of
the definition of M̃ beyond Γξ as necessary, we have
MA t
t ≤ Γξ
−rt
M̃t = e Xt =
Mξ
t > Γξ .
Recall that ϕ is extended to (x0 , ∞) in such a way that limx↑∞ ϕ(x) = 0.
Therefore either s(∞) < ∞ and ν assigns infinite mass to all points z >
s(∞) = z ν , or s(∞) = ∞ and there exists a sequence an ↑ ∞ such that ν
assigns mass to any neighbourhood of an .
B = ξ < ∞,
Suppose, we are in the second case. If s(0) > −∞ then Hs(0)
B = ξ, Γ is strictly increasing at u = H B and
else ξ = ∞. On HaBn < Hs(0)
u
an
B
AΓH B = Han . Set
an
(
B ;
HaBn < Hs(0)
ΓHaB ,
n
(34)
Tn =
B ,
∞,
HaBn > Hs(0)
where the second line is redundant if s(0) = −∞. Then ATn = HaBn ∧ ξ
is such that M̃tTn := M̃t∧Tn = MAt ∧ξ∧HaB ≤ g(an ) and Tn is a reducing
n
sequence for M̃ .
Now suppose s(∞) < ∞ and ḡ = ∞. Choose an ↑ s(∞) such that ν
B
B we have
assigns mass to any neighbourhood of an . Then, on Hs(∞)
< Hs(0)
by the argument after Lemma 6.1 that ΓHaB ↑ ∞ almost surely, and the
n
argument proceeds as before with Tn given by (34) a reducing sequence.
Finally, suppose s(∞) < ∞ and ḡ < ∞. Then M is bounded by ḡ and M̃
is a martingale.
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E. Ekström: Department of Mathematics, Uppsala University, Box 480, SE751 06 Uppsala, SWEDEN
D. Hobson: Department of Statistics, University of Warwick, Coventry, CV4
7AL, UK